\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 148, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/148\hfil Dirichlet problem] {Dirichlet problem for a second order singular differential equation} \author[W. Zhou\hfil EJDE-2006/148\hfilneg] {Wenshu Zhou} \address{Wenshu Zhou \newline Department of Mathematics, Jilin University, Changchun 130012, China} \email{wolfzws@163.com} \thanks{Submitted July 31, 2006. Published December 5, 2006.} \thanks{Supported by grants 10626056 from Tianyuan Youth Foundation and 420010302318 \hfill\break\indent from Young Teachers Foundation of Jilin University} \subjclass[2000]{34B15} \keywords{Singular differential equation; positive solution; existence} \begin{abstract} This article concerns the existence of positive solutions to the Dirichlet problem for a second order singular differential equation. To prove existence, we use the classical method of elliptic regularization. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} We study the existence of positive solutions for the second order singular differential equation u''+\lambda\frac{u'}{t-1}-\gamma \frac{|u'|^2}{u}+f(t)=0, \quad 00$,$f(t) \in C^1[0,1]$and$f(t)>0$on$[0,1]$. It is well known that boundary value problems for singular ordinary differential equations arise in the field of gas dynamics, flow mechanics, theory of boundary layer, and so on. In recent years, singular second order ordinary differential equations with dependence on the first order derivative have been studied extensively, see for example \cite{a1, b4, j1, o1, o2, s1, t1, w1} and references therein where some general existence results were obtained. We point out that the case considered here is not in their considerations since it does not satisfy some sufficient conditions of those papers. Our considerations were motivated by the model, which arises in the studies of a degenerate parabolic equation (see for instance \cite{b1, b2,b3}), considered by Bertsch and Ughi \cite{b3} in which they studied \eqref{e1} with$\lambda=0$and$f\equiv1$and the boundary boundary conditions:$ u(1)=u'(0)=0$. By ordinary differential equation theories, they obtained a decreasing positive solution. However, it is easy to see from the boundary conditions \eqref{e2} that any positive solution to the Dirichlet problem for \eqref{e1} must not be decreasing. Recently, in \cite{z1} the authors studied the Dirichlet problem for \eqref{e1} with$\lambda=0$, and proved that if$\gamma>0$, then the problem has a positive solution$u$; moreover, if$\gamma>\frac{1}{2}$, then$u$satisfies also$u'(1)=u'(0)=0$. Note that the equation considered here is more general since it is also singular at$t=1$for$\lambda\neq0$. Thus the existence result obtained here is not a simple extension of \cite{b3, z1}. We say$u \in C^2(0,1) \cap C[0,1]$is a solution to the Dirichlet problem \eqref{e1}, \eqref{e2} if it is positive in$(0,1)$and satisfies \eqref{e1} and \eqref{e2}. The main purpose of this paper is to prove the following theorem. \begin{theorem}\label{thm1} Let$\lambda>-1, \gamma>\frac{1}{2}(1+\lambda)$,$f(t) \in C^1[0,1]$and$f>0$on$[0,1]$. Then the Dirichlet problem \eqref{e1}, \eqref{e2} has a solution$u$. Moreover,$u$satisfies$u'(1)=0$. If in addition we assume that$\lambda$is non-negative, then$u$satisfies also$u'(0)=0$. \end{theorem} \section{Proof of Theorem \ref{thm1}} We will use the classical method of elliptic regularization to prove Theorem \ref{thm1}. For this, we consider the following regularized problem: \begin{gather*} u''+\lambda\frac{u'}{t-1-\varepsilon^{1/2}}-\gamma \frac{|u'|^2\mathop{\rm sgn}{}_\varepsilon(u)}{I_\varepsilon(u)}+f(t)=0, \quad 0-1$ and using \eqref{e5}, we have \begin{align*} 0 \geqslant& \varepsilon^{\lambda/2}u_\varepsilon'(1) -\frac{A\varepsilon^{(1+\lambda)/2}}{1+\lambda} \\ \geqslant& (1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t) -\frac{ A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda}\\ \geqslant&(1+\varepsilon^{1/2})^{\lambda}u_\varepsilon'(0) - \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda}\\ \geqslant & - \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda},\quad t \in [0,1], \end{align*} and hence $\frac{A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda} \geqslant (1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t) \geqslant -\frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda}, \quad t \in [0,1].$ This completes the proof of Lemma \ref{lem2.1}. \end{proof} Obviously, we have \begin{gather} -u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+ \gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon}-\min_{[0,1]}f \geqslant 0,\quad t \in (0,1),\label{e6}\\ -u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+ \gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon} - \max_{[0,1]}f\leqslant 0,\quad t \in (0,1).\label{e7} \end{gather} To obtain uniform bounds of $u_\varepsilon$, the following comparison theorem will be proved to be very useful. \begin{proposition}\label{prop1} Let $u_i \in C^2(0,1)\cap C[0,1]$ and $u_i>0$ on $[0,1] (i=1,2)$. If $u_2 \geqslant u_1$ for $t=0,1$, and \begin{gather} -u_2''-\eta\frac{u_2'}{t-1-\rho}+\varrho\frac{|u_2'|^2}{u_2}-\theta \geqslant 0,\quad t \in (0,1),\label{e8}\\ -u_1''-\eta\frac{u_1'}{t-1-\rho}+\varrho\frac{|u_1'|^2}{u_1}-\theta \leqslant 0,\quad t \in (0,1),\label{e9} \end{gather} where $\rho, \varrho, \theta>0, \eta \in \mathbb{R}$, then $u_2(t)\geqslant u_1(t)$, $t \in [0,1]$. \end{proposition} \begin{proof} From \eqref{e8} and \eqref{e9}, we have \begin{gather*} \Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)'' +\frac{\eta}{t-1-\rho}\Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)' \leqslant-\frac{\theta}{u_2^{\varrho}}, \quad (\varrho \neq 1)\\ \Big({\rm ln} (u_2)\Big)'' +\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_2)\Big)' \leqslant-\frac{\theta}{u_2}, \quad (\varrho =1) \end{gather*} and \begin{gather*} \Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)'' +\frac{\eta}{t-1-\rho}\Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)' \geqslant-\frac{\theta}{u_1^{\varrho}}, \quad (\varrho \neq 1) \\ \Big({\rm ln} (u_1)\Big)'' +\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_1)\Big)' \geqslant-\frac{\theta}{u_1}. \quad (\varrho =1) \end{gather*} Combining the above inequalities, we obtain w''+\frac{\eta}{t-1-\rho} w' \leqslant \theta \Big(\frac{1}{u_1^{\varrho}}-\frac{1}{u_2^{\varrho}}\Big),\quad 0\frac{1}{2}(1+\lambda)$, we find that \[ -v_{\varepsilon}''-\lambda\frac{v_{\varepsilon}'}{t-1-\varepsilon^{1/2}}+ \gamma \frac{|v_{\varepsilon}'|^2}{v_{\varepsilon}}-\max_{[0,1]}f \geqslant 0,\quad 00$ in $(0, 1)$, and $\lim_{t\to 1^-}u(t)=0$. Define $u(1)=0$. Thus $u$ is a solution to the Dirichlet problem \eqref{e1}, \eqref{e2}, and it follows from \eqref{e15} that $u'(1)=0$. It remains to show that for $\lambda \geqslant 0$, $u$ satisfies $u'(0)=0$. Let $h_{\varepsilon_j}=C(t+\varepsilon_j^{1/2})^2$, where $C \geqslant \max\Big\{1, \frac{\max_{[0,1]}f }{2(2\gamma -1)}\Big\}$. Noticing $\lambda \geqslant 0$ and $\gamma>\frac{1}{2}(1+\lambda)$, we have \begin{align*} &-h_{\varepsilon_j}''-\lambda\frac{h_{\varepsilon_j}'}{t-1-\varepsilon_j^{1/2}}+ \gamma \frac{|h_{\varepsilon_j}'|^2}{h_{\varepsilon_j}}-\max_{[0,1]}f \\ &=2C(2\gamma -1)-2C\lambda\frac{t+\varepsilon_j^{1/2}}{t-1-\varepsilon_j^{1/2}}-\max_{[0,1]}f\\ &\geqslant 2C(2\gamma -1)-\max_{[0,1]}f \geqslant 0,\quad 0